In that case, you really can't use any hosted search engine because they could be logging all your queries, the time you made them, and associating them with your IP and any other information they can get their hands on. Are you familiar with any search engine that is technically incapable of logging your IP???
You can use a privacy-focused VPN like Mullvad to conceal your identity from the search engine. In most markets having a credit/debit card requires personal information so there's nothing you can do if they don't provide a privacy-friendly payment alternative.
Anyway, GP's question was simply:
"Can I use Kagi without linking any real-world identifiers (including credit card details) to my search queries in any way?"
